Jim and Jenny want to see who can drink their milkshake the fastest. They each buy a supersize cup, which is 1600 millilitres. Below is a graph which describes Jim’s drinking rate from the time he begins to drink the milkshake. They decide to see who can drink exactly half of the milkshake first. Jenny finishes half of the milkshake in 34 seconds. Who finished half of their milkshake first? You must show your working.
So Jim and Jenny both bought the supersize cup, which holds 1600 millilitres. But we’re looking to find who drinks exactly half of the milkshake first. So we want to take half of 1600, which would be 800 millilitres. So we have to decide who drinks 800 millilitres, exactly half, of their milkshake first, Jim or Jenny. And Jenny finishes half of the milkshake in 34 seconds. So if Jim finishes in a smaller amount of time than 34 seconds, he would finish first. However, if he took longer than 34 seconds, then he would finish second. And we wanna know who finished their milkshake first.
So our goal will be to find the time at which Jim drank 800 millilitres of his milkshake. And from our graph, we know that the volume drunk will be equal to the area under the graph. And looking at our graph, on the 𝑥-axis, that represents time in seconds. And the 𝑦-axis represents the drinking rate, with millilitres per second.
So the area underneath this graph will actually be the volume that they drank. And we want to know when they drank exactly 800 millilitres, that time. We already know Jenny’s was 34 seconds. We need to find Jim’s. From at zero to 10 seconds, to find out how much Jim drank during this time, we need to find the area underneath the graph between zero and 10 seconds. So let’s split this into a rectangle and a triangle.
The area of a rectangle is length times width. And the rectangle has 10 seconds and goes up to 15 millilitres per second. So this is a 10-by-15 rectangle. And 10 times 15 is 150 millilitres. For the area of a triangle, we have to take one-half times the base times the height. The base will be 10 seconds, and our height will be found by taking 40 minus 15, which is 25. So our area would be one-half times 10 times 25.
One-half times 10 is five, and five times 25 is 125. So the area of the triangle is 125 millilitres. So the area from zero to 10 seconds, the area underneath this part of the graph, will be 150 millilitres plus 125 millilitres, which is 275 millilitres. So the first 10 seconds of drinking this milkshake, Jim drank 275 millilitres. And that’s less than the 800 millilitres that we’re looking for, so we must keep going.
Let’s see how much he drank from 10 to 20 seconds. It’s a rectangle. And the base will be 10, and the height will be 40. So to find the area, we’ll take 10 times 40, so we get 400 millilitres. So during this 10-second time frame, he drank 400 millilitres. So if we want the total of what he’s drank so far, so the amount drunk after 20 seconds would be the 400 millilitres that we just found plus the 275 millilitres that he drank in the first 10 seconds, which is equal to 675 millilitres. This is still less than half of the milkshake, so we must keep going.
From 20 to 35 seconds, we have another rectangle with a base of 15 and a height of 10. And 15 times 10 is 150 millilitres. So during this 15-second time frame, he drank 150 millilitres. So how much did he drink altogether so far?
Well, after 35 seconds, he will have drank the 150 millilitres plus what he drank before, the 675 millilitres. And that’s 825 millilitres. Well, this is actually larger than the 800 millilitres. So he drank more than half of it after the 35 seconds. So he must have finished his milkshake some time between 20 seconds and 35 seconds.
We know that, at 20 seconds, he drank a total of 675. So how much more did he need to go to get to exactly half of the milkshake? Well, we can take 800 minus 675, and we know how much more he would need. He would need to drink 125 more millilitres. Well, how much time would it take for him to drink 125 millilitres? We can add that to our 20 seconds and we would know exactly when he finished half of the milkshake. So we are unsure of the exact time.
However, no matter where the time is between 20 and 35 seconds, the height of this rectangle is going to stay 10. And instead of drinking 150 millilitres, we wanna know when will be he have drunk 125 millilitres.
So using the area formula of length times width equalling the area, we can say length times width equals the area. So 10 times the time — we can call it 𝑡 — equals the area of 125 millilitres. So we can solve for 𝑡 by dividing by 10. And we find that 𝑡 is equal to 12.5 seconds. So we need to take the 20 seconds plus 12.5 seconds. And this will let us know when Jim finished half of the milkshake.
So taking 20 seconds plus 12.5 seconds, we get 32.5 seconds. So Jim took 32.5 seconds, while Jenny took 34 seconds. So Jim was actually faster at drinking exactly half of the milkshake. So who finished half of their milkshake first? It would be Jim.